Dirichlet forms and degenerate elliptic operators

Details Dirichlet forms and degenerate elliptic operators Dirichlet forms and degenerate elliptic operators

2006-01-14

arxiv.org/abs/math/0601349v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) \mapsto d(A ;B)\in[0,\infty]$ over pairs of measurable subsets of $\Ri^d$. Then \[ |(\phi_A,S_t\phi_B)|\leq e^{-d(A;B)^2(4t)^{-1}}\|\phi_A\|_2\|\phi_B\|_2 \] for all $t>0$ and all $\phi_A\in L_2(A)$, $\phi_B\in L_2(B)$. Moreover $S_tL_2(A)\subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=\infty$ where $A^c$ denotes the complement of $A$.

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